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The reader might be left wondering why we should be interested in James Gregory's proof of this result. Priority--being the first to publish a proof of the result--certainly isn't an issue here. The first published proof of the Pappus-Guldin theorem appeared more than 20 years before Gregory's *GPU*. It's also not the case that Gregory's proof is more elegant than those presented by his predecessors. Roccha's proof is only about a paragraph long, and once some concepts from the theory of indivisibles are mapped into modern ideas of calculus it does not differ significantly from the proofs found in most calculus books. Perhaps the best apologist for Gregory's work is Gregory himself. Near the end of the *GPU* he describes his purpose:

These particular problems selected by me, besides certain problems now first solved by me, were found to be very difficult and of great importance among geometers. But indeed Archimedes' entire work "On The Sphere and the Cylinder" is easily demonstrated from Proposition 3 in the manner of Proposition 46 and a few of the following propositions. His book "On Conoids and Spheroids" and all of Luca Valerius' work is demonstrated from Proposition 21; all of the work of Guldin, Ioannis della Faille, and Andreas Tacquet is demonstrated from Proposition 35 and a few of the subsequent propositions.

In other words, Gregory's contribution is abstraction. For Gregory, the Pappus-Guldin Theorem (and quite a few other results) are easy consequences of a broader geometrical perspective--that is, a perspective involving ratios between the trunk, the cylinder, and the solid of revolution.

In making these abstract connections, he is part of a trend that dominated much of 17th century mathematical thinking. Whereas Greek mathematicians and early 17th Century mathematicians had found solutions to specific problems, later mathematicians worked with entire classes of objects and looked at connections between them. As we know, the culmination of this drive to abstraction would be the Fundamental Theorem of Calculus, which would cement the connection between integration and differentiation and almost completely trivialize area and volume computations. For mathematicians today, it is easy to underestimate the revolutionary impact that the Fundamental Theorem of Calculus had on mathematical thinking. But the contrast with the fate of the Pappus-Guldin Theorem helps put it into perspective. As Pappus himself recognized more than 1700 years ago, his theorem gives a truly remarkable and quite surprising connection between volumes of revolution, areas, and centers of gravity. But now it appears essentially as a margin note in calculus texts--completely usurped in its importance by the Fundamental Theorem of Calculus.

So too with James Gregory's contribution to the Pappus-Guldin Theorem. His ratios between right cylindrical figures, solids of revolution, and the trunk give an important and noteworthy connection between the two completely familiar and yet seemingly different ways of constructing solids out of 2-dimensional figures. But the connections and abstractions he saw were geometrical and not analytical. While the fundamental importance of Gregory's "trunk" may seem obvious to someone trained to think in terms of Euclidean proportion theory, what Gregory failed to recognize was that the importance of classical geometry arguments in mathematics was on the decline. Even as Gregory was proving his abstract geometrical connections, Newton and others were already successfully employing the algebraic techniques and methods of analysis that are still in the ascendance in mathematics today.

Andrew Leahy (Knox College), "James Gregory and the Pappus-Guldin Theorem - Conclusion," *Convergence* (February 2010), DOI:10.4169/loci003262